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$X\sim R(0,1)$ and $Y\sim R(0,1)$ (X and Y are uniformly distributed on the interval $[0,1])$

I need to find the density function of W when $W=X\cdot Y$

Can anyone help me?

Mathe
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  • I like to approach such problems by finding the cumulative distribution function for $W$. – hardmath Dec 25 '17 at 12:56
  • All right. How would you do that? I must be honest to say, that this kind of problem is a little bit difficult for me. In my probability class, it is the hardest problem we solve. So I don't really have a clue how to start it. – Mathe Dec 25 '17 at 13:08
  • In the $x,y$ plane, the condition $xy\le t$ cuts out a certain region from the unit square. Its area is the desired cumulative distribution function evaluated at $t$. So you want a formula for that, and then to differentiate that formula. Pictures help a lot with this problem – kimchi lover Dec 25 '17 at 13:12
  • The joint distribution of $X,Y$ on $[0,1]\times [0,1]$ is uniform, so Pr$(W \le w)$ is the area of the unit square "bounded" by the curve $xy = w$. – hardmath Dec 25 '17 at 13:12
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    You should add to your question that $X$ and $Y$ are independent. Without that information the problem cannot be solved. – drhab Dec 25 '17 at 13:13
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    A google search shows this popular post: https://math.stackexchange.com/questions/659254/product-distribution-of-two-uniform-distribution-what-about-3-or-more. – StubbornAtom Dec 25 '17 at 13:48

1 Answers1

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Hint (preassuming that $X$ and $Y$ are independent):

For $w\in(0,1)$ we have: $$F_{W}(w)=\int\int1_{(-\infty,w]}(xy)f_X(x)f_Y(y)dxdy=\int^w_0\int^{w/y}_0dxdy$$

Work this out and find PDF $f_W(w)$ as derivative of $F_W(w)$.

drhab
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